Self-InductanceInductance

•Consider a solenoid L, connect it to a battery•Area A, lengthl, N turns

•What happens as you close the switch?•Lenz’s law – loop resists change in magnetic field•Magnetic field is caused by the current•“Inductor” resists change in current

+–

0NIB

0

1B

NIA

B

L

d

dt

E 1B

dN

dt

20N IAd

dt

20N A dI

dt

dIL

dtE

20N A

L

A

l

Inductors•An inductor in a circuit is denoted by this symbol:•An inductor satisfies the formula:•L is the inductance

•Measured in Henrys (H)

dIL

dtE

1 H 1 V s/A

L

Kirchoff’s rules for Inductors:•Assign currents to every path, as usual•Kirchoff’s first law is unchanged•The voltage change for an inductor is L (dI/dt)

•Negative if with the current•Positive if against the current

•In steady state (dI/dt = 0) an inductor is a wire

+–

L

I

Energy in Inductors•Is the battery doing work on the inductor?

I V P

+–

L

dIIL

dt

•Integral of power is work done on the inductor

U dtPdI

IL dtdt

L IdI 212 LI k

•It makes sense to say there is no energy in inductor with no current21

2U LI

•Energy density inside a solenoid?2 2

0

2

N AIU

0NI

B

20L N A

Uu

A

2 20

22

N I

2

02

Bu

•Just like with electric fields, we can associate the energy with the magnetic fields, not the current carrying wires

RL Circuits•An RL circuit has resistors and inductors•Suppose initial current I0 before you open the switch •What happens after you open the switch?•Use Kirchoff’s Law on loop•Integrate both sides of the equation

L

+–

I

R

0dI

Ldt

RIdI RI

dt L

dI Rdt

I L

dI Rdt

I L

ln constantRt

IL

Rt LI e

0tI I e

L R

RL Circuits (2)•Where did the energy in the inductor go?•How much power was fed to the resistor?

L

+–

R

2RIP

0tI I e

L R 2 20

tR LRI e

•Integrate to get total energy dissipated

0

RU dt

P 2 20

0

Rt LRI e dt

20 02

Rt LLRI e

R

2102 I L

2102LU LI

•It went to the resistor

•Powering up an inductor:•Similar calculation

1 tI eR

E

L =

4.0

mH

I R0tI I e

L R An inductor with inductance 4.0 mH is discharging through a resistor of resistance R. If, in 1.2 ms, it dissipates half its energy, what is R?

210 02U LI 21

2U LI 102 U 21

04 LI

2 2102fI I

0

10.707

2

I

I

0

0.707tIe

I

ln 0.707 0.347t

0.347

t 31.2 10 s

0.00346 s0.347

LR

.00400 H

.00346 s 1.16 R

Sample Problem

Inductors in Series•For inductors in series, the inductors have the same current•Their EMF’s add

1

dIL

dtE 2

dIL

dt 1 2

dIL L

dt

1 2L L L

•For inductors in parallel, the inductors have the same EMF but different currents

L1

L2

11

dIL

dtE

22

dIL

dtE

1 2dI dIdI

dt dt dt

1 2L L

E E

L

E

1 2

1 1 1

L L L

Inductors in Parallel

L1L2

Parallel and Series - Formulas

Capacitor Resistor Inductor

Series

Parallel

Fundamental Formula

1 2R R R

1 2

1 1 1

R R R 1 2C C C

1 2

1 1 1

C C C

1 2L L L

1 2

1 1 1

L L L

QV

C V IR L

dIL

dtE

Materials Inside Inductors•For capacitors, we gained significantly by putting materials inside•Can we gain any benefit by putting something in inductors?•It gives you something to wrap around

Can materials increase the inductance?•Most materials have negligible magnetic properties•A few materials, like iron are ferromagnetic

•They can enhance inductance enormously•Many inductors (and similar devices, like transformers) have iron cores

•We will ignore this

Symbol for iron core inductor:•We won’t make this distinction

Mutual Inductance•Consider two solenoids sharing the same volumeWhat happens as you close the switch?•Current flows in one coil•But Lenz’s Law wants mag. flux constant•Compensating current flows in other coil•Allows you to transfer power without circuits being actually connected•It works even better if source is AC from generator

+–

I1 I2

LC Circuits•Inductor (L) and Capacitor (C)•Let the battery charge up the capacitorNow flip the switch•Current flows from capacitor through inductor•Kirchoff’s Loop law gives:•Extra equation for capacitors:

+–

C

L

Q

0Q C V C EI

0Q

C

dIL

dt

dQI

dt dI

Q CLdt

d dQ

CLdt dt

2

2

1d QQ

dt CL

•What function, when you take two deriva-tives, gives the same things with a minus sign?•This problem is identical to harmonicoscillator problem

cos

sin

Q t

Q t

0 cosQ Q t

LC Circuits (2)•Substitute it in, see if it works

C

L

Q

I

0 cosQ Q t

0 sindQ

Q tdt

2

202

cosd Q

Q tdt

2

2

1d QQ

dt CL

20 0

1cos cosQ t Q t

CL

1

CL

•Let’s find the energy in the capacitor and the inductor

dQI

dt 0 sinQ t

2

2C

QU

C

220 cos

2C

QU t

C

212LU LI 2 2 21

02 sinLQ t

2

20 sin2L

QU t

C

20

2C L

QU U

C

Energy sloshes back and forth

Frequencies and Angular Frequencies•The quantity is called the angular frequency•The period is the time T you have to wait for it to repeat•The frequency f is how many times per second it repeats

2T

1

CL 0 cosQ Q t T

1f T2 f

WFDD broadcasts at 88.5 FM, that is, at a frequency of 88.5 MHz. If they generate this with an inductor with L = 1.00H,

what capacitance should they use?

2 f 6 12 88.5 10 s 8 15.56 10 s

2 1LC 2

1C

L

28 1 6

1

5.56 10 s 10 H

3.23 pFC

RLC Circuits•Resistor (R), Inductor (L), and Capacitor (C)•Let the battery charge up the capacitorNow flip the switch•Current flows from capacitor through inductor•Kirchoff’s Loop law gives:•Extra equation for capacitors:

+–

C

L

Q

I

0Q dI

L RIC dt

dQI

dt

2

20

Q dQ d QR L

C dt dt

•This equation is hard to solve, but not impossible•It is identical to damped, harmonic oscillator

20 cosRt LQ Q e t

R

2

2

1

4

R

LC L